NDA Three Dimensional Geometry questions with solutions
Lines, planes, spheres and direction cosines in space. 48 questions from our 2016–2025 NDA pattern papers, from the Class 12 Mathematics syllabus. Each has the answer and a full solution.
NDA 2025
- The point on the z-axis which is equidistant from (1,5,7) and (5,1,-4) is
- The distance of the plane 2x-3y+6z=14 from the origin is
- The lines with direction ratios (1,2,2) and (2,k,-3) are perpendicular if k is equal to
- The angle θ between the planes x+y+z=1 and x-y+z=2 satisfies
- The radius of the sphere with centre at the origin which touches the plane 2x+3y+6z=35 is
- The direction cosines of a line can be (1/2,1/2,1/2). 2. The direction cosines l,m,n of a line satisfy l^2+m^2+n^2=1.
NDA 2024
- The perpendicular distance of the point (2,-3,6) from the x-axis is
- If l,m,n are its direction cosines, then l+m+n equals
- The equation of the plane passing through (1,2,3) and having normal direction ratios 2,-1,4 is
- The acute angle it makes with the x-axis is
- The foot of the perpendicular drawn from the point (3,4,5) to the xy-plane is
- Consider the points A(1,2,3), B(2,4,6) and C(3,6,9). 1. A, B, C are collinear. 2. B divides AC in the ratio 2:1.
- The points A(1,2,3), B(3,5,4) and C(5,8,λ) are collinear if λ equals
NDA 2023
- The mid-point of the segment joining (2,-4,6) and (4,0,-2) is
- A line makes equal angles with the three coordinate axes. Its direction cosines can be
- The equation of the plane which contains the x-axis and passes through the point (0,2,3) is
- The planes x+2y+2z=3 and 2x+y-2z=5 are
- The sphere with centre (1,2,3) that touches the xy-plane has the equation
- 1/2,1/2,1/2 can be the direction cosines of a line. 2. Lines with direction ratios (1,2,3) and (3,0,-1) are perpendicular.
NDA 2022
NDA 2021
NDA 2020
- The distance of the point (3,-4,12) from the z-axis is
- The equation of the plane through the point (1,2,3) and perpendicular to the line with direction ratios 2,-1,4 is
- The cosine of the angle between two lines with direction ratios (1,2,2) and (2,-3,6) is
- A line can have direction cosines (1/2,1/2,1/2). 2. A line can have direction cosines ((1)/(√2),(1)/(√2),0).
- The foot of the perpendicular from the origin to the plane 2x+y+2z=18 is
- The shortest distance between the lines vec r=(1,2,3)+t(1,1,0) and vec r=(3,3,6)+s(0,1,1) is
NDA 2019
- The distance of the point (2,-3,5) from the y-axis is
- If θ is the acute angle between the planes x+2y+2z=5 and 4x-3z=2, then cos θ equals
- A line is perpendicular to each of two lines whose direction ratios are (1,2,2) and (2,1,-2). Its direction cosines are
- The sine of the angle between the line with direction ratios (1,2,2) and the plane 2x-y+2z=5 is
- The planes ax+by+cz=d1 and ax+by+cz=d2 with d1≠ d2 are parallel. 2. Two lines with direction ratios (1,2,3) and (2,4,6) must be the same line.
NDA 2018
- The perpendicular distance of the point (3,4,12) from the x-axis is
- An acute-angled line makes angles 60^° and 45^° with the x-axis and y-axis. The angle it makes with the z-axis is
- The plane z=3 cuts the sphere x^2+y^2+z^2=25 in a circle of radius
- The plane passing through (1,1,1) and perpendicular to the line with direction ratios 2,3,6 is
- The line (x-2)/(1)=(y-1)/(2)=(z-3)/(2) meets the plane x+y+z=11 at the point
NDA 2017
- A line makes an angle of 60^° with each of the x-axis and y-axis and an acute angle γ with the z-axis. Then γ equals
- The plane through the point P(3,4,12) perpendicular to the line OP (O being the origin) is at a distance of what from the origin?
- The plane through (1,-1,2) perpendicular to the vector 3hat i-2hat k has the equation
- The acute angle between the planes x+y=3 and y+z=5 is
- A line equally inclined to the three coordinate axes has direction cosines ±(1)/(√3),±(1)/(√3),±(1)/(√3). 2. The distance of the point (1,2,3) from…
- The cosine of the angle which the line (x-1)/(2)=(y+2)/(3)=(z-3)/(6) makes with the z-axis is
NDA 2016
Other Mathematics topics
Vector Algebra 60Complex Numbers and Quadratic Equations 35Binomial Theorem 16Sequences and Series 36Quadratic Equations 27Permutations and Combinations 25Statistics 35Probability 27Matrices 37Determinants 13Sets, Relations and Functions 39Binary Number System 11Logarithms 11Inverse Trigonometric Functions 22Trigonometric Functions and Identities 49Trigonometric Equations 16Properties of Triangles 18Heights and Distances 12Straight Lines 43Circles 21Conic Sections 28Limits 23Continuity and Differentiability 16Application of Derivatives 34Derivatives 33Integrals 16Definite Integrals 23Application of Integrals 12Differential Equations 30